Some Effective Methods in the Openness of Loci for Cohen-Macaulay and Gorenstein Properties

Rossi, Fabio; Spangher, Walter

doi:10.1007/978-1-4612-0441-1_29

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…This context ("polynomial regular sequence + Noether position") appears frequently in several approaches related to effectivity problems in Computer Algebra, even in the positive dimensional case (see for instance [20], [11], [21]). At this point it is quite natural to look for properties of R -bases of S (degree bounds, algorithms to compute them, etc.…”

Section: Lemma 22mentioning

confidence: 99%

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On the degrees of bases of free modulesover a polynomial ring

Almeida¹,

D'Alfonso²,

Solernó³

1999

Math Z

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Section: Lemma 22mentioning

confidence: 99%

“…This situation appears frequently in problems related to effective elimination theory (see [20], [12], [6], [1], [14], [11]). In this context it is natural to ask about quantitative properties of bases of the module S .…”

Section: Introductionmentioning

confidence: 99%

On the degrees of bases of free modulesover a polynomial ring

Almeida¹,

D'Alfonso²,

Solernó³

1999

Math Z

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“…More precisely, let f 1 , ..., f n−r ∈ k [x 1 , ..., x n ] be a regular sequence in a polynomial ring over a perfect field k. Suppose that the variables are in Noether position, in other words the canonical morphism R := k [x 1 , ..., x r ] → S := k [x 1 , ..., x n ] / (f 1 , ..., f n−r ) is injective and integral. It is well known that under these hypotheses the ring S is an R−free module of finite rank (see [15,Corollary 18.17], [22,Lemma 3.3.1], [42]). This situation appears in a very natural way when one looks for effective solutions of polynomial systems.…”

Section: The Coordinates Of the Vectors V I Are Polynomials Of Degreementioning

confidence: 99%

Computing bases of complete intersection rings in Noether position

Almeida¹,

Blaum²,

D'Alfonso³

et al. 2001

Journal of Pure and Applied Algebra

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Let k be an effective infinite perfect field, k[x1, ..., xn] the polynomial ring in n variables and F ∈ k[x1, ..., xn] M ×M a square polynomial matrix verifying F 2 = F. Suppose that the entries of F are polynomials given by a straight line program of size L and their total degrees are bounded by an integer D. We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time (nL) O(1) (M D) O(n). By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely: let f1, ..., fn−r ∈ k[x1, ..., xn] be a regular sequence of polynomials given by a slp of size , whose degrees are bounded by d. Let R := k[x1, ..., xr] and S := k[x1, ..., xn]/ (f1, ..., fn−r) such that S is integral over R; we show that there exists an algorithm running in time O (n) d O(n 2) which computes a basis of S over R. Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k[x1, ..., xn]−module given by a presentation matrix, and in the affirmative case it computes a basis.

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“…shows the following statement:Let V C IK" be an ideal-theoretic complete intersection algebraic variety of dimension r. Let us assume that the variables X\,... ,X" are in Noether position with respect to V. Then, K[T] is a free K[Xi,...,Xr]-module of finite rank. For a proof of this statement see[69,4] or[31, Lemma 3.3.1],…”

mentioning

confidence: 99%

On the intrinsic complexity of the arithmetic Nullstellensatz

Hägele

Morais

Pardo

et al. 2000

Journal of Pure and Applied Algebra

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We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorith mic procedure computing the polynomials and constants occurring in a Bezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again poly nomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function 7ls(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteris tic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

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Some Effective Methods in the Openness of Loci for Cohen-Macaulay and Gorenstein Properties

Cited by 4 publications

References 8 publications

On the degrees of bases of free modulesover a polynomial ring

On the degrees of bases of free modulesover a polynomial ring

Computing bases of complete intersection rings in Noether position

On the intrinsic complexity of the arithmetic Nullstellensatz

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